1. Chapter 4 Continuous Random Variables and Probability Distributions
4.1 - Probability Density Functions
4.2 - Cumulative Distribution Functions and
Expected Values
4.3 - The Normal Distribution
4.4 - The Exponential and Gamma Distributions
4.5 - Other Continuous Distributions
4.6 - Probability Plots

2. ~ The Normal Distribution ~ (a.k.a. “The Bell Curve”)X
Johann Carl Friedrich Gauss
standard deviation
X ~ N(μ, σ) σ
Symmetric, unimodal
Models many (but not all) natural systems
Mathematical properties make it useful to work with
mean μ

3. Standard Normal Distribution
Z ~ N(0, 1)
SPECIAL CASE
Total Area = 1 1 Z
The cumulative distribution function (cdf) is denoted by (z). It is not expressible in explicit, closed form, but is tabulated, and computable in R via the command pnorm.

4. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Total Area = 1 1 Z
1.2
“z-score”

5. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Use the included table.
Total Area = 1 1 Z
1.2
“z-score”

6. Lecture Notes Appendix…

8. Standard Normal Distribution
Example
Standard Normal Distribution
Z ~ N(0, 1)
Find (1.2) = P(Z  1.2).
Use the included table.
Use R:
> pnorm(1.2)
[1]
Total Area = 1 1
P(Z > 1.2) Z
1.2
“z-score”
Note: Because this is a continuous distribution, P(Z = 1.2) = 0, so there is no difference between P(Z > 1.2) and P(Z  1.2), etc.

9. Standard Normal Distribution
Z ~ N(0, 1) μ σ
X ~ N(μ, σ) 1 Z
Why be concerned about this, when most “bell curves” don’t have mean = 0, and standard deviation = 1?
Any normal distribution can be transformed to the standard normal distribution via a simple change of variable.

10. Random Variable
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
Year 2010
X ~ N(25.4, 1.5)
μ = 25.4
σ = 1.5
27.2

11. Random Variable POPULATION Example X ~ N(25.4, 1.5)
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
The x-score = 27.2 must first be transformed to a corresponding z-score.
Year 2010
X ~ N(25.4, 1.5)
σ = 1.5
μ = 25.4
μ = 25.4
μ =
27.2 33

12. Random Variable
X = Age at first birth
POPULATION
Example
Question: What proportion of the population had their first child before the age of 27.2 years old?
P(X < 27.2) = ?
P(Z < 1.2) =
Year 2010
X ~ N(25.4, 1.5)
σ = 1.5
Using R:
> pnorm(27.2, 25.4, 1.5)
[1]
μ = 25.4
μ =
27.2 33

13. Standard Normal Distribution
Z ~ N(0, 1) 1 Z
What symmetric interval about the mean 0 contains 95% of the population values?
That is…

14. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
0.95
0.025
0.025 Z
-z.025 = ?
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?
That is…

15. Lecture Notes Appendix…

17. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
Use R:
> qnorm(.025)
[1]
> qnorm(.975)
[1]
0.95
0.025
0.025 Z
-z.025 = -1.96
-z.025 = ?
“.025 critical values”
+z.025 = +1.96
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?

18. Standard Normal Distribution
Z ~ N(0, 1)
X ~ N(25.4, 1.5)
X ~ N(μ, σ)
What symmetric interval about the mean age of 25.4 contains 95% of the population values?
22.46  X  yrs
> areas = c(.025, .975)
> qnorm(areas, 25.4, 1.5)
[1]
0.95
0.025
0.025 Z
-z.025 = -1.96
-z.025 = ?
“.025 critical values”
+z.025 = +1.96
+z.025 = ?
What symmetric interval about the mean 0 contains 95% of the population values?

19. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
0.90
0.05
0.05 Z
-z.05 = ?
+z.05 = ?
Similarly…
What symmetric interval about the mean 0 contains 90% of the population values?

20. …so average 1.64 and 1.65
0.95  average of and …

21. Standard Normal Distribution
Z ~ N(0, 1)
Use the included table.
Use R:
> qnorm(.05)
[1]
> qnorm(.95)
[1]
0.90
0.05
0.05 Z
-z.05 = ?
-z.05 =
+z.05 =
+z.05 = ?
“.05 critical values”
Similarly…
What symmetric interval about the mean 0 contains 90% of the population values?

22. Standard Normal Distribution
Z ~ N(0, 1)
In general….
1 – 
0.90
0.05
 / 2
 / 2
0.05 Z
-z / 2
-z.05 = ?
-z.05 =
+z.05 =
+z / 2
+z.05 = ?
“ / 2 critical values”
“.05 critical values”
Similarly…
What symmetric interval about the mean 0 contains 100(1 – )% of the population values?

23. Normal Approximation to the Binomial Distribution
continuous
discrete
Normal Approximation to the Binomial Distribution
Suppose a certain outcome exists in a population, with constant probability .
We will randomly select a random sample of n individuals, so that the binary “Success vs. Failure” outcome of any individual is independent of the binary outcome of any other individual, i.e., n Bernoulli trials (e.g., coin tosses).
Discrete random variable
X = # Successes in sample
(0, 1, 2, 3, …,, n)
Discrete random variable
X = # Successes in sample
(0, 1, 2, 3, …,, n)
P(Success) = 
P(Failure) = 1 – 
Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with “probability function”
p(x) = , x = 0, 1, 2, …, n.

24. > dbinom(10, 100, .2)
[1]
Area

25. > pbinom(10, 100, .2)
[1]
Area

30. “Sampling Distribution” of
Therefore, if…
X ~ Bin(n, ) with n  15 and n (1 – )  15,
then…
That is…
“Sampling Distribution” of

31. Classical Continuous Probability Distributions
Normal distribution
Log-Normal ~ X is not normally distributed (e.g., skewed), but Y = “logarithm of X” is normally distributed
Student’s t-distribution ~ Similar to normal distr, more flexible
F-distribution ~ Used when comparing multiple group means
Chi-squared distribution ~ Used extensively in categorical data analysis
Others for specialized applications ~ Gamma, Beta, Weibull…